In certain energy configurations our method is a noticeable improvement of the standard Rabi transitions strategy and leads to ” active tunnelling control”, as illustrated by simulations

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FLATNESS BASED TRAJECTORY GENERATION OF QUANTUM SYSTEMS

Pierre Rouchon

Ecole des Mines de Paris, Centre Automatique et Syst`emes, 60 Bd Saint-Michel, 75272 Paris cedex 06

FRANCE. Email: [email protected] NOLCOS 2004

Abstract: A two-states quantum system with one control is proved to be flat. This provides a simple procedure to design smooth open-loop controls that steer in finite time from one eigen-state to the other one. A three-states quantum system with one control is not flat in general. Following the Rabi oscillations used by physicists to control stimulated transition, we associate to this system an averaged control system where the number of controls is increased and where flatness-based motion planning techniques can be used. This allows to steer directly from one eigen-state to any other one without using an additional intermediate eigen-state. In certain energy configurations our method is a noticeable improvement of the standard Rabi transitions strategy and leads to ” active tunnelling control”, as illustrated by simulations. This method can be extended without major difficulties to higher dimensional cases.

Keywords: Quantum oscillators, controllability, averaging, motion planning, flatness based control, trajectory generation.

1. INTRODUCTION

The control of quantum system is an active sub- ject of fundamental and practical importance in chemistry (see, e.g., (Levis et al., 2001)). It ap- pears that the controllability of quantum systems described by a finite number of states is quite well understood (see, e.g. (Ramakrishna et al., 1995; Turinici, 2000b; Turinici, 2000a)). How- ever there is still a need to have simple control design method. In (Vettori, 2002; Ferranteet al., 2002; Mirrahimi and Rouchon, 2004a; Mir- rahimi and Rouchon, 2004b) Lyapounov based methods are proposed. This paper proposes flatness based methods (Fliess et al., 1995; Fliess et al., 1999) for steering from a pure state to another pure state. The main advantage of such method is its computational simplicity and its simple interpretation in physical terms.

We consider quantum systems described by the following Schr¨odinger equation

ı~ ψ˙ = (H0+uH1)ψ (1) where ψ is the complex probability amplitude vector (belongs to an Hilbert space), H0 is the free Hamiltonian and H1 is the Hamiltonian associated to the scalar controlu(corresponding to a classical electric field generated by a laser). We focus here on the following motion planning prob- lem: for two pure states,ψa andψb of free energy EbandEb(H0ψa=EaψaandH0ψb=Ebψb), find an open-loop control [0, T] 3 t 7→ u(t) steering the state ψ form ψa at t = 0 to the state ψb at t=T >0.

We first consider a two-state system (ψ∈C²) that is proved to be flat for almost all HamiltoniansH0

and H1. The flat output admits a clear physical

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interpretation when the Bloch sphere model is used. The second class of systems we consider has three states ψ ∈ C³ and a special structure of H₀ and H₁ as illustrated on figure 1. Such structure renders one particular transition between two eigen-states not so easy (two-photons transition).

Such transition requires, when Rabi oscillations are used, to pass via the remaining third eigen- state. Our method avoids this intermediate step.

The steering control is then obtained by consid- ering, under the classical weak field approximation, the average dynamics that admits then two control variables. Flatness-based motion planning techniques provide easily the steering control for the averaged dynamics (see (Fliess et al., 1995) where a similar method is used to control the Kapitsa pendulum). Simulations show that such approximate steering control remains quite attrac- tive since the amplitude of the control does not need to be very small to reach the final state.

Preliminary version of these results can be found in (Rouchon, 2002).

2. TWO STATES SYSTEMS

Consider now a two states system. Its wave func- tionψbelongs toC². The Schr¨odinger equation

ı~ψ˙= (H0+uH1)ψ

involves the 2×2 Hermitean matricesH₀andH₁: H0=

µ−E/2 0

0 E/2

¶

, H1= µh1 b

b^∗ h2

¶

with E, h1, h2 ∈ R and b ∈ C^∗. With ψ = (a1, a2)∈C², the density matrix is defined by

ρ=|φi hφ|=

µ|a1|² a^∗₁a2

a1a^∗₂ |a2|²

¶ . In terms of Pauli matrices

σx= µ0 1

1 0

¶

, σy= µ0 −ı

ı 0

¶

, σz = µ1 0

0 −1

¶ , it reads

ρ= 1 +λσx+µσy+νσz

with S = (λ, µ, ν) ∈ S². This corresponds to a classical change of coordinates (Bloch sphere model, see (Cohen-Tannoudjiet al., 1977; Abragam, 1961)) where the meaningless absolute phase is removed: φ= (a1, a2)∈ C²/{0} and ˜φ(˜a1,˜a2) ∈ C²/{0} are the probabilities amplitudes of the same physical state if and only if exists θ ∈ S¹ such thata= exp(ıθ)˜a. Notice thatS∈S²comes from |a1|²+|a2|² = 1. In the Bloch coordinates the dynamics reads

S˙ =S∧(ω0B0+u

~B1) whereω0=E/~and

B0=



0 0 1



, B1=



−2<(b) 2=(b) h1−h2



.

Set τ = ω0t, ⁰ = d/dτ and v = ^kB_ω¹^k

0~ u the new control. The dynamics becomes

S⁰ =S∧(B0+vJ) where J is the unitary vector _kB¹

1kB1. Denote by α ∈]0, π[ the angle between B0 and J and consider the ortho-normal frame (I,J,K) with K=B₀∧J/sinαandI=J∧K. SetS=xI+ yJ +zK ((x, y, z)∈R³ with x²+y²+z² = 1).

Then the dynamics reads x⁰ =−z(cosα+u) y⁰ =zsinα

z⁰ =x(cosα+u)−ysinα

sinceB0= sinαI+cosαJ. Notice thatx²+y²+z² is invariant. The restriction of the dynamics onS² is flat withy as flat output:

z=y⁰/sinα, x=± q

1−y²−(y⁰)²/sin²α.

Let us now find a smooth control [0, T] 3 t 7→

u(t), u(0) = 0 and u(T) = 0, steering from

−E/2 to +E/2. When u= 0, the state of energy

−E/2 corresponds, in the Bloch-coordinates, to (x, y, z) = (−sinα,−cosα,0) and the state of energy +E/2 to (x, y, z) = (sinα,cosα,0).

Assume to simplify that α = π/2 (a similar construction exists for over values of α). Set p(τ) = (1−τ)τ²(2−τ)². Simple computations show that the functionf(τ) = 1−p²(τ)−(p⁰)²(τ) is non negative on [0,2], reaches 0 only for τ= 1 with f⁰⁰(1) > 0. Thus the function g : R 7→ R defined by

g(τ) =











−1 ifτ <0

−p

f(τ) ifτ ∈[0,1]

pf(τ) ifτ ∈[1,2]

1 ifτ >2.

isC². The control

v(τ) =−g⁰(τ)/p⁰(τ)

is well defined even for τ around 1. It steers the system from (−1,0,0) at τ = 0 to (1,0,0) at τ = 2. The steering trajectory is

x(τ) =g(τ), y(τ) =p(τ), z(τ) =p⁰(τ).

The interest of the above computations relies on the fact that the control u is smooth. This is not the case for standard steering control of

±1/2 spin systems (Abragam, 1961): u is then piecewise constant and discontinuous; the steering trajectory is made of Larmor precessions over finite time interval.

3. THREE STATES SYSTEM

Consider now a three states system with three energy levels E1 < E2 < E3 associated to the

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tunnel transition via active control

classical Rabi transition

Fig. 1. A three states system: steering fromE1 to E2without reachingE3via active tunnelling control.

physical case illustrated on figure 1. It corresponds to the reduced model of a 1Dparticule in the po- tentialV(q) admitting two minima with bounded statesψ1 and ψ2 of low energy E1 and E2 sepa- rated by a potential barrier and a third bounded state ψ3 of energy E3 passing over the barrier.

The supports of the ψi, i = 1,2,3 are roughly sketched by the dashed horizontal lines. The three states model is a finite dimensional approximation of (p=ı/~_∂q^∂ )

ı~ψ˙ = (p²/2m+V(q)−uq)ψ

then entries (1,2) and (2,1) in H1 are negligible.

We have the following dynamics ı~α˙1 = (E1+uhqi₁)α1+uhqi₁₃α3

ı~α˙2 = (E2+uhqi₂)α2+uhqi₂₃α3

ı~α˙3 = (E3+uhqi₃)α3+uhqi^∗₃₁α1+uhqi^∗₃₂α2

where hqi_i=

Z

|ψi|²q dq, hqi_ij= Z

ψ_i^∗ψjq dq andψ=α1ψ1+α2ψ2+α3ψ3withai∈C. Change the phase as follows fori= 1,2,3:

αi= exp µ

−ıE3/~t− Z _t

0

u(s)hqi₃/~ds

¶ ai. Then

ı˙a1 = (ω13+ue1)a1+ub1a3

ı˙a2 = (ω23+ue2)a2+ub2a3

ı˙a3 =ub^∗₁a1+ub^∗₂a2

(2) whereω13= (E1−E3)/~andω23= (E2−E3)/~

are the Bohr frequencies, ei = (hqi_i − hqi₃)/~, b1=hqi₁₃/~ andb2=hqi₂₃/~.

Finding explicit open-loop control [0, T] 3 t 7→

u(t) steering from the pure state of energy E1, a = (1,0,0), to the pure state E2, a = (0,1,0) without passing via the intermediate state E3 is not so obvious. We propose here an open-loop design mixing standard perturbation techniques (see, e.g., (Messiah, 1962)) and flatness based steering methods.

Assume that the controluis small,|ubi|,|uej| ¿ ω13, ω23 and varies slowly (time constant much smaller than T23 = 2π/ω23 and T13 = 2π/ω13

the Bohr periods). Set b1 = r1exp(ıθ1), b2 = r₂exp(ıθ₂) withr_i>0 andθ_i real. Set

u= 2v1(t)

r1 cos(ω13t) +2v2(t)

r2 cos(ω23t) with v1 and v2 small amplitude. Then classical averaging techniques show that, when ω13/ω23 is irrational, the solutions of (2) are close to the solution of the average system

˙

x1 =v1x3

˙

x2 =v2x3

˙

x3 =−v1x1−v2x2

(3) where

a1= exp(ı(θ1−ω13t))x1

a2= exp(ı(θ2−ω23t))x2

a3=ıx3.

When one of the v_i is constant and the other one is zero we recover the classical Rabi oscillations (Messiah, 1962). They can be used to steer, in a first step, the state from energyE1to energy E3 with v1 constant 6= 0 and v2 = 0 and then, in a second step, to steer the system from E3 to E2 with v1= 0 and v2 constant 6= 0. We will see that we can mix these two steps to steer directly the system from E1 to E2 without reaching the energyE3.

Up to phase shifts that are not important from physical reasons, we have to findv1andv2steering the state x from (1,0,0) to (0,1,0). Thus we can suppose that the components of x remain real during the motion (since u must be real,v₁ andv2 are also real). Conservation of probability means that I = x²₁+x²₂ +x²₃ is invariant and equal to 1 and we have (with the positive branch x3=p

1−x²₁−x²₂):

˙ x1=v1

q

1−x²₁−x²₂, x˙2=v2

q

1−x²₁−x²₂ withx1andx2as flat output. Take any increasing smooth bijections7→σ(s) from [0,1] to [0,1] with

σ(0) = 0, σ(1) = 1, dⁱσ

dsⁱ(0) =dⁱσ dsⁱ(1) = 0 for i = 1,2,3. Set x1 = 1−σ(t/T) and x2 = σ(t/T). Then the control

v1(t) = −σ⁰(t/T) Tp

2σ(t/T)(1−σ(t/T)) v2(t) = σ⁰(t/T)

Tp

2σ(t/T)(1−σ(t/T))

is well defined fort∈[0, T], is smooth and satisfies vi(0) =vi(T) = 0, i= 1,2. Moreover it steers the average state xfrom (1,0,0) at t= 0 to (0,1,0) at t = T. Notice that when T À T13, T23, we automatically satisfy the averaging assumptions.

The simulations here below are based on model (2) withσa polynomial of degree 7. We have consid- ered three different cases:

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−0.4

−0.2 0 0.2 0.4

Parameters: b 1=1, b

2=1, e 1=−1, e

2=1, ω₁₃=1, ω₂₃=1.5708, T/T 23=5

v1 v2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−0.5 0 0.5 1

Control u= v

1(t) cos(ω₁₃ t) + v 2(t) cos(ω₂₃ t)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 0.5 1

Probabilities

|a1|²

|a2|²

|a3|²

Fig. 2. Active tunneling from E1 to E2 without reachingE3> E1, E2; the standard case.

• the standard case of figure 2: thebiandeiare closed to one; the ratio of the Bohr frequency is irrational.

• the resonant case of figure 3: the ratio of the Bohr frequency is rationalω23= 2ω13.

• the ill conditioned case of figure 4: the system is close to a non controllable one;ω23≈ω13. This case requires larger transition timesT.

4. GENERALIZATION

Such control designs can be extended to quantum oscillators with more than 3 states. Such an exten- sion relies on non-resonance assumptions, averaging techniques and the fact that the controllable part of the averaged system is flat.

Assume that H0/~ in (1) is the diagonal matrix diag(ωk), with ωk ∈ R for k = 1, ..., n, n being the dimension of the system. Denote bybk,l ∈C the entries of H1/~. The goal is to steer the system from the first pure state (1,0, ...,0) to the last one (0, ...,0,1). Assume that exists a path (l1, ..., lσ)∈ {1, ..., n}^σ of lengthσ≥2 such that, lk 6= lr for k 6= r, l1 = 1, lσ = n and for each r∈ {1, ..., σ−1}, blr,lr+1 6= 0. Set

u=

σ−1X

r=1

ur

³

e^ı(ω^lr^−ω^lr+1^)t+e^−ı(ω^lr^−ω^lr+1^)t´ whereurare new controls. Withψ= exp(−ıH0t/~)φ, (1) becomes

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−0.5 0 0.5

2=1, e 1=−1, e

2=1, ω₁₃=1, ω₂₃=2, T/T 23=5

v1 v2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−1

−0.5 0 0.5

Control u= v

1(t) cos(ω₁₃ t) + v 2(t) cos(ω₂₃ t)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 0.5 1

Probabilities

|a1|²

|a2|²

|a3|²

Fig. 3. Active tunneling from E1 to E2 without reachingE3> E1, E2; the resonant caseE3− E2= 2(E3−E1).

0 2 4 6 8 10 12 14 16 18 20

−0.1

−0.05 0 0.05 0.1

2=1, e 1=−1, e

2=1, ω₁₃=1, ω₂₃=1.1, T/T 23=20

v1 v2

0 2 4 6 8 10 12 14 16 18 20

−0.1 0 0.1 0.2 0.3

Control u= v 1(t) cos(ω

13 t) + v 2(t) cos(ω

23 t)

0 2 4 6 8 10 12 14 16 18 20

0 0.5 1

Probabilities

|a1|²

|a2|²

|a₃|²

Fig. 4. Active tunneling from E1 to E2 without reachingE3> E1, E2; the ill conditioned case E3−E2≈(E3−E1.

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ıφ˙k =

σ−1X

r=1

Xn

s=1

urbk,sφse^ı(ω^k^−ω^s^)t...

...

³

e^ı(ω^lr^−ω^lr+1^)t+ +e^−ı(ω^lr^−ω^lr+1^)t

´ . Assume that |ωk −ωs| = |ωlr −ωlr+1|, implies (k, s) = (lr, lr+1) or (s, k) = (lr, lr+1). For ur

small enough we can use the standard rotating wave approximation (called also secular approximation) and consider the following approximate average dynamics where the relative phases (the coherences) between the state components are lost:

ıφ˙k= 0 fork6∈{l1, ..., lσ} ıφ˙l1 =u1bl1,l2φl2

...

ıφ˙lr =urblr,lr−1φlr−1+ur+1blr,lr+1φlr+1

...

ıφ˙lσ =uσ−1blσ,lσ−1φlσ−1

The components that do not belong to the transition path remain constant. Set blr,lr+1 = crexp(ıθk) with cr > 0 and θr ∈ R. Set vr = crurand consider the following relative change of phases:

φl1 =x1

φl2 =ıe^−ıθ¹x2

...

φlr = (ı)^r−1e^−ı(θ¹^+...+θ^r−1⁾xr

...

φlσ = (ı)^σ−1e^−ı(θ¹^+...+θ^σ−1⁾xσ.

In the variablesx= (x1, ..., xσ), the approximate dynamics reads:

˙

x1=v1x2

˙

x2=−v1x1+v2x3

...

˙

xσ−1=−vσ−2xσ−2+vσ−1xσ

˙

xσ−2=−vσ−1xσ−1.

We consider now only real values forx(remember that the coherences are lost due to the rotating wave approximation). Notice that x²₁ +... +x²_σ is an invariant (conservation of probability). The above system lives thus on the unit sphere of R^σ. It is obviously controllable since it hasσ−1 independent controls (v1, ..., vσ−1). It is trivially flat withxas flat output.

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